Representation with a highest weight vector

A linear representation (cf. Representation of a Lie algebra) $ \rho $ of a finite-dimensional semi-simple split Lie algebra $ \mathfrak g $ over a field $ k $ of characteristic zero with a split Cartan subalgebra $ \mathfrak t $, having the following properties.

1) In the space $ V $ of $ \rho $ there is a cyclic vector $ v $( i.e. $ V $ is the smallest $ \mathfrak g $- invariant subspace containing $ v $).

2) $ \rho ( h) v = \lambda ( h) v $ for all $ h \in \mathfrak t $, where $ \lambda $ is some fixed linear form on $ \mathfrak t $ with values in $ k $.

3) If $ \alpha _ {1} \dots \alpha _ {r} $ is a system of simple roots, defined by a lexicographical order on the set $ \Delta $ of all roots of $ \mathfrak g $ relative to $ \mathfrak t $( cf. Root system), and if $ e _ {\alpha _ {i} } , \mathfrak t _ {\alpha _ {i} } , h _ {\alpha _ {i} } $ are the vectors from the Chevalley basis of $ \mathfrak g $ corresponding to $ \alpha _ {i} $, $ i = 1 \dots r $, then $ \rho ( e _ {\alpha _ {i} } ) ( v) = 0 $ for all $ i = 1 \dots r $. Thus, $ \lambda $ is a weight relative to the restriction of $ \rho $ to $ \mathfrak t $( cf. Weight of a representation of a Lie algebra); it is called a highest weight. The space $ V $ is called a cyclic $ \mathfrak g $- module with highest weight $ \lambda $ and generator $ v $, and $ v $ is called a highest weight vector.

There exists for every linear form $ \lambda $ on $ \mathfrak t $ a unique, up to equivalence, irreducible representation $ \rho _ \lambda $ of $ \mathfrak g $ with highest weight $ \lambda $. The $ \mathfrak g $- module $ V ( \lambda ) $ determined by $ \rho _ \lambda $ is a direct sum of weight subspaces relative to the restriction of $ \rho _ \lambda $ to $ \mathfrak t $. Their weights have the form

$$ \lambda - \sum _ {i = 1 } ^ { r } n _ {i} \alpha _ {i} , $$

where the $ n _ {i} $ are non-negative integers. The weight subspace $ V _ \mu ( \lambda ) $ of weight $ \mu $ is finite-dimensional, spanned over $ k $ by vectors of the form

$$ ( \rho _ \lambda ( f _ {\alpha _ {i _ {1} } } ) \dots \rho _ \lambda ( f _ {\alpha _ {i _ {s} } } ) ) ( v ) , $$

and for any $ h \in \mathfrak t $ the restriction of $ \rho _ \lambda ( h) $ to $ V _ \mu ( \lambda ) $ is the operator of scalar multiplication by $ \mu ( h) $. The space $ V _ \lambda ( \lambda ) $ is one-dimensional; the weight $ \lambda $ is the only highest weight of $ \rho _ \lambda $ and can be characterized as the unique weight of the $ \mathfrak t $- module $ V ( \lambda ) $ such that any other weight has the form

$$ \lambda - \sum _ {i = 1 } ^ { r } n _ {i} \alpha _ {i} , $$

where the $ n _ {i} $ are non-negative integers.

A representation $ \rho _ \lambda $ is finite-dimensional if and only if $ \lambda $ is a dominant linear form on $ \mathfrak t $, i.e. $ \lambda ( h _ {\alpha _ {i} } ) $ is a non-negative integer for $ i = 1 \dots r $. Every irreducible finite-dimensional linear representation of $ \mathfrak g $ has the form $ \rho _ \lambda $ for some dominant linear form $ \lambda $ on $ \mathfrak t $( hence all such representations are classified, up to equivalence, by the dominant linear forms on $ \mathfrak t $). The set of all weights of a finite-dimensional representation $ \rho _ \lambda $ relative to $ \mathfrak t $ is invariant relative to the Weyl group of $ \mathfrak g $( regarded as a group of linear transformations of $ \mathfrak t $), and if weights $ \mu $ and $ \gamma $ belong to one orbit of the Weyl group, then the dimensions of the spaces $ V _ \mu ( \lambda ) $ and $ V _ \gamma ( \lambda ) $ are equal. For every weight $ \mu $ and every root $ \alpha \in \Delta $ the number $ \mu ( h _ \alpha ) $ is an integer; if, moreover, $ \mu + \alpha $ is also a weight, then

$$ \rho ( e _ \alpha ) ( V _ \mu ( \lambda )) \neq 0 $$

(here $ h _ \alpha $ is the element in $ \mathfrak t $ corresponding to $ \alpha $ and $ e _ \alpha $ is the root vector of $ \alpha $).

References

[1]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[2]	, Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Secr. Math. Univ. Paris (1955)
[3]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)
[4]	E. Cartan, "Les tenseurs irréductibles et les groupes linéaires simples et semi-simples" Bull. Sci. Math. , 49 (1925) pp. 130–152
[5]	Harish-Chandra, "On some applications of the universal enveloping algebra of a semisimple Lie algebra" Trans. Amer. Math. Soc. , 70 (1951) pp. 28–96